Do you have an aftermarket fan assembly? I had one of these fail on me after nine months of use, and it failed with your exact symptoms. It was also wired in reverse, but that had been fixed.

The fan controller is much more than a simple relay and resistor. It pulses the motor (the pulses that you saw) to start it, and reads the electrical signature of the motor during these pulses. If the signature is wrong (say, your cheap aftermarket fan motor magnets have weakened), the fan fails to start, and another attempt is made in a few seconds. If the motor bumps during this attempt, give it some help (spin the hub with your hand) to see if it will start. If the motor is going bad but not all the way gone and the controller is good, this may start it up.

Note that testing whether the fan runs with 12V applied is not a valid test of the controller. Mine ran just fine (but not as strong as new) with 12V applied, but the new OEM assembly did cure my problem.